It is well-known that metric \(f(R)\)  theories of gravity have a scalar-tensor representation, which can be achieved by means of a Legendre transformation. We will present here only the final result, as the process of deriving the ST representation can be found in many papers, for example, in [1]}:
\(S[g_{\mu \nu }, \Phi , \chi ] = \frac{1}{2\kappa ^2}\int _\Omega d^4x \sqrt{-g}[\phi R(g) - V_{f(R)}(\Phi )] + S_\text{matter}[g_{\mu \nu }, \chi ],\) 
